Double-Slit Interference

Double-slit interference

Concept: Two narrow slits next to each other do not make two bright stripes. Waves from the two slits overlap and add up crest-to-crest in some directions (bright) and crest-to-trough in others (dark), giving many equally spaced fringes. Each slit is also a single slit, so the fringes fade inside a wider single-slit envelope.

What the four steps show

• 1 · Setup — slit separation d, slit width a, wavelength λ, screen distance L. Fringe spacing: Δy = λL/d.
• 2 · Two sets of waves — each slit sends out its own waves. Green lines: directions where crest meets crest (bright). Grey dotted: crest meets trough (dark).
• 3 · Bright or dark? — ray 2 has a shorter path to P. Path difference Δ = d·sinθ. A whole number of wavelengths: in phase, bright. Half a wavelength: opposite phase, dark. P sweeps from the centre through the first dark fringe to the first bright fringe.
• 4 · Pattern — I(y) = I₀ cos²(πdy/λL) · sinc²(πay/λL). The dashed grey curve is the single-slit envelope.

Key formulas

• Bright: d·sinθ = mλ   Dark: d·sinθ = (m + ½)λ
• Fringe spacing: Δy ≈ λL/d   Envelope zero: y = λL/a

Sources

Hecht, E. (2017). Optics (5th ed.), Pearson, ch. 9–10.
Halliday, Resnick & Walker, Fundamentals of Physics, “Interference” chapter.